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the Technology Interface/Fall 2007 Cartwright DETERMINING THE EFFECTIVE OR RMS VOLTAGE OF VARIOUS WAVEFORMS WITHOUT CALCULUS by Kenneth V. Cartwright, Ph.D. [email protected] School of Sciences and Technology College of The Bahamas Abstract: The rms (root-mean-square) value of various waveforms is determined without the use of calculus, which should be of benefit to the Technology student. These waveforms include the half-wave rectified sinusoid, the quarter-wave rectified sinusoid, a waveform consisting of a dc voltage added to a sinusoid, the sum of two sinusoids, which may be periodic or aperiodic, a triangular waveform, and a pulse train. Furthermore, in order to compute the rms value, the mean of the square of a voltage must be determined over an observation time. A discussion of the importance of using the correct observation time is also provided. I. Introduction Several authors, e.g. [1] and [2], have shown that the effective or root-mean-square (rms) voltage of a sinusoid can be obtained without the use of calculus. In this paper, we show that this can also be done for other voltage waveforms, although this does not seem to be widely recognized. These methods all use the definition of the rms voltage of a waveform v(t ) , which is found simply by squaring that waveform, taking the mean (or average) of that squared waveform and then computing the square root, as given in equation (1): Vrms = mean[v(t ) * v(t )] = area under the curve of [v(t ) * v(t )] , observation length (1) where the observation length is the period (or integer multiple of it) for periodic signals, and is as long as possible for aperiodic signals. More will be said about this in Section II C. This definition is, of course, a purely mathematical one. The physical interpretation of the rms value has already been covered by many authors, including Rico [1]. Hence, in this 1 the Technology Interface/Fall 2007 Cartwright paper, we will be considering (1), without any attempt to repeat its physical interpretation, which remains the same as it was for the purely sinusoidal case. II. Analytical Determination of the RMS Value of Some Selected Waveforms In this section, we will show how the rms value of the following waveforms can be determined without calculus: (i) (ii) (iii) (iv) (v) (vi) Half-wave rectified sinusoid Quarter-wave rectified sinusoid A DC voltage added to a sinusoid The sum of two sinusoids, when the sum waveform is periodic or aperiodic A triangular waveform, and A pulse train. A. RMS Value of a Half-wave Rectified Sinusoid A graph of a positive half-wave rectified sinusoid is shown in Fig. 1. For this waveform, we give two different methods for finding its rms value. In Method I, the half-wave rectified waveform is written as the sum of a sinusoid and a full-wave rectified sinusoid. In Method II, we use a piecewise mathematical equation for the waveform. 2 the Technology Interface/Fall 2007 Cartwright v(t) (V) Vm 0 0 T/2 Time (s) T Fig. 1. Graph of a positive half-wave rectified sine-wave. Method I The equation for a positive half-wave rectified sinusoid can be written as v(t ) = Vm sin(ωt + θ ) / 2 + Vm sin(ωt + θ ) / 2. (2) Note that the second term in (2) is simply a full-wave rectified sinusoid. Squaring (2) gives v(t ) * v(t ) = Vm2 sin 2 (ωt + θ ) / 4 + Vm sin(ωt + θ ) / 4 + 2Vm sin(ωt + θ ) Vm sin(ωt + θ ) , which can also be written as 2 v 2 (t ) = Vm2 sin 2 (ωt + θ ) / 2 + 2Vm sin(ωt + θ ) Vm sin(ωt + θ ) . (3) Now we need to take the mean of (3), which consists of the sum of two terms. Fortunately, the mean of a sum is the sum of the means, i.e. 3 the Technology Interface/Fall 2007 mean(a + b) = mean(a ) + mean(b). In Cartwright this case, a = Vm2 sin 2 (ωt + θ ) and b = 2Vm sin(ωt + θ ) Vm sin(ωt + θ ) . Note the mean of the second term in (3) is zero, i.e. mean(b) = 0. To see this, we sketch b as a function of the angle (ωt + θ ) , as shown in Fig.2. Waveform of b 2Vm*Vm 0 -2Vm*Vm 0 90 180 Angle (deg) 270 360 Fig. 2. Graph of b, the second term in (3). This clearly shows that the mean(b) = 0. Note that for every positive value of b in Fig. 2, there is a corresponding negative value. Hence, the average over a complete cycle is zero. (Or, by symmetry, it is clear that the negative area under the curve for the second halfcycle has the same magnitude as the positive area under the curve for the first half cycle. Therefore, the total area over a complete cycle is zero, and because the mean is simply the area over a complete cycle, divided by the period, the mean has to be zero). On the other hand, the mean of a is given by mean(Vm2 sin 2 (ωt + θ ) / 2) = mean(Vm2 / 4 − Vm2 / 4 cos(2ωt + 2θ )), where we have used the well-known trigonometric identity 4 (4) the Technology Interface/Fall 2007 Cartwright sin 2 ( x) = (1 − cos(2 x)) / 2 . (5) Equation (4) further simplifies to mean(a) = mean(Vm2 / 4) − mean(Vm2 / 4 cos(2ωt + 2θ )). (6) The first term on the right-hand-side of (6) is simply the mean of a constant which of course is just that constant. On the other hand, the second term on the right-hand-side of (6) is the mean of a sinusoid, which is well-known to be zero, as pointed out by Rico [1]. Hence, mean(a ) = Vm2 / 4, which then of course makes the mean of (3) also this value. Finally, we need to take the square root of the mean of (3) to arrive at our conclusion: the rms value of a positive half-wave rectified sinusoid is Vm2 / 4 = Vm / 2, which is also the answer given by calculus. In the above, we assumed that we had a positive half-wave rectified sinusoid. For a negative half-wave rectified sinusoid, the proof is similar, except its equation is given by v(t ) = Vm sin(ωt + θ ) / 2 − Vm sin(ωt + θ ) / 2. Method II The equation for the positive half-wave rectified sine-wave shown in Fig. 1 can also be written in piecewise fashion as ⎛ 2π v(t ) = Vm sin ⎜ ⎝ T ⎞ t⎟ ⎠ for 0 ≤ t ≤ T 2 (7) T for ≤ t ≤ T . 2 =0 Squaring (7) gives ⎛ 2π v 2 (t ) = Vm2 sin 2 ⎜ ⎝ T ⎞ t⎟ ⎠ for 0 ≤ t ≤ T 2 (8) T for ≤ t ≤ T . 2 =0 A graph of this squared waveform is shown in Fig. 3. Furthermore, using (5), (8) can be written as a constant minus a cosine wave, as given below in (9). 5 the Technology Interface/Fall 2007 Cartwright v 2 (t ) = Vm2 Vm2 ⎛ 4π cos ⎜ − 2 2 ⎝ T ⎞ t⎟ ⎠ for 0 ≤ t ≤ T 2 (9) T for ≤ t ≤ T . 2 =0 v2(t) (V2) Vm*Vm 0 0 T/2 Time (s) T Fig. 3. Plot of the square of the half-wave rectified sine-wave of Fig.1. Inspection of (9) shows that the squared waveform can be written as the difference of two waveforms, i.e., v 2 (t ) = a(t ) − b(t ) , (10) where Vm2 a (t ) = 2 for 0 ≤ t ≤ =0 for and 6 T 2 T ≤t ≤T 2 the Technology Interface/Fall 2007 b(t ) = Cartwright Vm2 ⎛ 4π cos ⎜ 2 ⎝ T ⎞ t⎟ ⎠ for 0 ≤ t ≤ =0 for T 2 T ≤ t ≤ T. 2 These latter two waveforms are shown in Fig. 4(a) and Fig. 4(b). Now we need to find the mean of (10), which is ( ) ( ) ( ) ( ) mean v 2 (t ) = mean a 2 (t ) − b 2 (t ) = mean a 2 (t ) − mean b 2 (t ) . (11) Fortunately, the means on the right hand side of (11) are easily determined. Using the fact that the mean is simply the area under the curve over a period, divided by the period, as given in (1), we find mean a 2 (t ) = Vm2 / 4 and mean b 2 (t ) = 0 . Hence, from (11), ( ) ( ) ( ) mean v (t ) = V / 4 and so the rms value of the waveform in Fig. 1 is Vm / 2 , as 2 2 m determined earlier with Method I. a(t) (V2) Vm*Vm/2 0 0 T/2 Time (s) T Fig. 4(a). Piecewise constant component of the square of a half-wave rectified sinewave. Note that the mean of this waveform over a complete period is Vm2 / 4 . 7 the Technology Interface/Fall 2007 Cartwright b(t) (V2) Vm*Vm/2 0 -Vm*Vm/2 0 T/8 T/4 3T/8 T/2 Time (s) 5T/8 3T/4 7T/8 T Fig. 4(b). Piecewise sinusoidal component of the square of a half-wave rectified sinewave. Note that the mean of this waveform over a complete period is zero. B. RMS Value of a Quarter-Wave Rectified Sine-wave The quarter-wave rectified sine-wave shown in Fig. 5 is typically found in silicon controlled rectifier (SCR) circuits. Fortunately, the rms value of this waveform can be found with Method II, as described above in Section II-A. Indeed, the equation for the quarter-wave rectified sine-wave shown in Fig. 5 can be written in piecewise fashion as ⎛ 2π v(t ) = Vm sin ⎜ ⎝ T ⎞ t⎟ ⎠ =0 T T ≤t≤ 4 2 T T for 0 ≤ t ≤ and ≤ t ≤ T . 4 2 for (12) Squaring (12) gives ⎛ 2π v 2 (t ) = Vm2 sin 2 ⎜ ⎝ T =0 ⎞ t⎟ ⎠ T T ≤t ≤ 4 2 T T for 0 ≤ t ≤ and ≤ t ≤ T . 4 2 for (13) A graph of this squared waveform is shown in Fig. 6. Furthermore, using (5), (13) can be written as a constant minus a cosine wave, as given below in (14). 8 the Technology Interface/Fall 2007 Cartwright v(t) (V) Vm 0 0 T/4 T/2 Time (s) 3T/4 T Fig. 5. Graph of a positive quarter-wave rectified sine-wave, which is useful in siliconcontrolled rectifier circuits. v2(t) (V2) Vm*Vm 0 0 T/4 T/2 Time (s) 3T/4 T Fig. 6. Graph of the square of the quarter-wave rectified sine-wave. 9 the Technology Interface/Fall 2007 v 2 (t ) = Vm2 Vm2 ⎛ 4π cos ⎜ − 2 2 ⎝ T Cartwright ⎞ t⎟ ⎠ T T ≤t≤ 4 2 T T for 0 ≤ t ≤ and ≤ t ≤ T . 4 2 for =0 (14) As noted before, (14) can be written as the difference of two waveforms, i.e., v 2 (t ) = a(t ) − b(t ) , (15) where a(t ) = Vm2 2 T T ≤t ≤ 4 2 T T for 0 ≤ t ≤ and ≤ t ≤ T 4 2 for =0 and Vm2 ⎛ 4π ⎞ b(t ) = t⎟ cos ⎜ 2 ⎝ T ⎠ T T ≤t ≤ 4 2 T T for 0 ≤ t ≤ and ≤ t ≤ T . 4 2 for =0 These latter two waveforms are shown in Fig. 7(a) and Fig. 7(b). Now, using (11), we need to find the mean of (15). Fortunately, the means on the right hand side of (11) are easily determined to be mean a 2 (t ) = Vm2 / 8 and mean b 2 (t ) = 0 . ( ( ) ) ( Hence, mean v 2 (t ) = Vm2 / 8 and so the rms value of the waveform in Fig. 5 is ) Vm , as 2 2 can also be verified with calculus. We wish to point out that the rms value of other waveforms that are found in SCR or Triac circuits can be found using Method II, e.g., the waveform shown in Fig. 8. 10 the Technology Interface/Fall 2007 Cartwright a(t) (V2) Vm*Vm/2 0 0 T/4 T/2 Time (s) 3T/4 T Fig. 7(a). Piecewise constant component of the square of a quarter-wave rectified sinewave. Note that the mean of this waveform over a complete period is Vm2 / 8 . b(t) (V2) Vm*Vm/2 0 -Vm*Vm/2 0 T/4 T/2 Time (s) 3T/4 T Fig. 7(b). Piecewise sinusoidal component of the square of a quarter-wave rectified sinewave. Note that the mean of this waveform over a complete period is zero. 11 the Technology Interface/Fall 2007 Cartwright v(t) (V) Vm 0 -Vm 0 T/4 T/2 Time (s) 3T/4 T Fig. 8. Another waveform that can be found in electronic circuits, and whose rms value can be found with Method II. C. RMS Value of DC and a Sinusoid We now turn our attention to the rms value of a waveform that is the sum of a DC voltage and a sinusoid. In some Technology textbooks (e.g. [2]), this result is just given without any justification. In this subsection, we show how we can derive this result without calculus. In this case, v(t ) = Vdc + Vm sin(ωt + θ ), where Vdc is the value of the DC voltage and Vm is the amplitude of the sinusoid. Squaring this gives 2 2 2 Vdc + 2VdcVm sin(ωt + θ ) + Vm sin (ωt + θ ). Taking the mean produces the mean of three terms. The mean of the first term is simply Vdc2 . The mean of the second term is zero since we are taking the mean of a sinusoid. The mean of the third term is mean(Vm2 / 2 − Vm2 / 2 cos(2ωt + 2θ )), where we have again used (5). Hence, the mean of this third term becomes Vm2 / 2. Therefore, mean(Vdc2 + 2VdcVm sin(ωt + θ ) + Vm2 sin 2 (ωt + θ )) = Vdc2 + Vm2 / 2. 12 (16) the Technology Interface/Fall 2007 Cartwright Taking the square root of (16) gives our desired result, i.e. the rms value of a DC voltage plus a sinusoid is given by Vrms = Vdc2 + Vm2 / 2. D. RMS Value of the Sum of Two Sinusoids In this subsection, we find the rms value of the sum of two sinusoids, i.e. we assume that v(t ) = v1 (t ) + v2 (t ) = Vm1 sin(ω1t + θ1 ) + Vm 2 sin(ω2t + θ 2 ). First, we square this waveform to get v 2 (t ) = Vm21 sin 2 (ω1t + θ1 ) + Vm21 sin 2 (ω2t + θ 2 ) + 2Vm1Vm 2 sin(ω1t + θ ) sin(ω2t + θ 2 ). Using (5) and the trigonometric identity sin( x) sin( y ) = cos( x − y ) / 2 − cos( x + y ) / 2 , we can write v 2 (t ) = Vm21 / 2 − Vm21 / 2 cos(2ω1t + 2θ1 ) + Vm22 / 2 − Vm22 / 2 cos(2ω2t + 2θ 2 ) + Vm1Vm 2 cos[(ω1 − ω2 )t + θ1 − θ 2 ] + Vm1Vm 2 cos[(ω1 + ω2 )t + θ1 + θ 2 )]. (17) Inspection of (17) shows that the square consists of v 2 (t ) = Vm21 / 2 + Vm22 / 2 + sum of sinusoids. Hence, when we properly take the mean of (17), the mean of the sum of sinusoids will be zero and so the rms value will be Vrms = Vm21 / 2 + Vm22 / 2. By properly taking the mean of (17), we mean that the observation length (time) over which the mean of v 2 (t ) is computed must be carefully considered. There are three different cases to consider: (i) one sinusoid is an harmonic of the other, i.e. ω2 = N ω1 , where N is a positive integer, (ii) the sinusoids are not harmonically related, but nonetheless the sum is periodic, and (iii) the sum of the sinusoids is not periodic. In the first case, the observation time is simply the period of the sinusoid with the lowest frequency. In the second case, the observation time is the period (or integer multiple) of the sum of the sinusoids contained in v 2 (t ) and not the period of each individual sinusoid. The reader should recall that the period of the sum of sinusoids might be quite longer than the period of the individual sinusoid that has the longest period. For example, suppose so that v 2 (t ) has frequencies 2ω1 = 6π rad/s, v(t ) = sin(3π t ) + sin(6π / 5t ) ; 2ω2 = 12π / 5 rad/s, ω1 − ω2 = 9π / 5 rad/s and ω1 + ω2 = 21π / 5 rad/s. The corresponding period of each of these is T1 = 1/ 3 s, T2 = 5 / 6 s, T3 = 10 / 9 s and T4 = 10 / 21 s, respectfully. Hence, in this case, the period of the sum is given by Tsum = 10 / 3 s, as this is the minimum time necessary to ensure that all the sinusoids have gone through complete cycles. Indeed, for this example, Tsum = 10 / 3s = 10T1 = 4T2 = 3T3 = 7T4 : that is, the first sinusoid completes ten cycles, the second completes four, the third sinusoid completes three, and the fourth completes seven cycles. 13 the Technology Interface/Fall 2007 Cartwright Fig. 9(a) shows a PSpice plot of v (t ) = sin(3π t ) + sin(6π / 5t ) , and Fig. 9(b) shows the square of that same waveform. Clearly, the period of v 2 (t ) is Tsum = 10 / 3 s, as expected. Therefore, the time over which the mean of v 2 (t ) is computed should be integer multiples of 10/3 s. This fact is verified in Fig. 9(c), which shows a PSpice plot of the mean of v 2 (t ) as a function of observation length (time). From the graph, it is clearly seen that PSpice rightly determines the correct mean value of unity at integer multiples of 10/3 s, and also at other observations lengths. However, to determine these latter observation lengths requires calculus: hence, we will limit ourselves to calculating the mean over the period of the sum, or integer multiples of this. Finally, in the third case, the period of the sum of two sinusoids might be infinite and so the mean computation has to be taken at specific values of time, or can only be wellapproximated by computing the mean over a “long” period of time. For example, see Fig. 10(a) which shows a PSpice plot of v(t ) = sin(2π t ) + sin(2 2π t ) . Note that this waveform is not periodic because the ratio of the frequencies of its sine waves is an irrational number. Also, see Fig. 10(b) which shows the square of this waveform, which also is not periodic. Furthermore, Fig. 10(c) shows the mean of the squared waveform as a function of observation length (time). Clearly, the correct determination of the mean value of unity for this aperiodic waveform depends upon the mean being taken over specific times. However, to calculate these specific times requires the use of calculus. On the other hand, as can be seen from Fig. 10(c), the longer the observation time to calculate the mean, the more accurate the calculation becomes. Therefore, we need not worry about determining the exact times over which the mean should be calculated; rather, we just allow the observation time to be a lot longer than the period of the lowest frequency sine wave in the squared waveform. We end this subsection by pointing out that this method of finding the rms value of a waveform which is the sum of two sinusoids can easily be generalized to a waveform which is the sum of N sinusoids, where N is any positive integer. 14 the Technology Interface/Fall 2007 Cartwright 2.0V (6.6667,0.000) (3.3333,0.000) 1.0V 0V -1.0V -2.0V 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 10s V(R1:2) Time Fig. 9(a). Plot of v(t ) = sin(3π t ) + sin(6π t / 5) . Note that the period is 10/3 s. 4.0 (3.3333,0.0000) (6.6667,0.0000) 3.0 2.0 1.0 0 0s 1s V(R1:2)* V(R1:2) 2s 3s 4s 5s 6s 7s 8s 9s 10s Time Fig. 9(b). Plot of v 2 (t ) = ( sin(3π t ) + sin(6π t / 5) ) . Note that the period is 10/3 s. 2 2.0 (10.000,1.0000) 1.5 (6.6667,1.0000) (3.3333,1.0000) 1.0 0.5 0 0s 1s 2s AVG(V(R1:2)*V(R1:2)) 3s 4s 5s 6s 7s 8s 9s Time 2 Fig. 9(c). Plot of mean ⎡⎣v 2 (t ) ⎤⎦ = mean ⎡( sin(3π t ) + sin(6π t / 5) ) ⎤ as a function of ⎣ ⎦ observation length (time) of the mean calculation. 15 10s the Technology Interface/Fall 2007 Cartwright 2.0V 1.0V 0V -1.0V -2.0V 0s 2s 4s 6s 8s 10s 12s 14s 16s V(R1:2) Time Fig. 10(a). Plot of v(t ) = sin(2π t ) + sin(2 2π t ) . Note the lack of periodicity. 4.0 3.0 2.0 1.0 0 0s 2s V(R1:2)* V(R1:2) 4s 6s 8s 10s 12s 14s 16s Time ( ) 2 Fig. 10(b). Plot of v 2 (t ) = sin(2π t ) + sin(2 2π t ) . Note the lack of periodicity. 3.0 2.0 1.0 0 0s 2s AVG(V(R1:2)*V(R1:2)) 4s 6s 8s 10s 12s 14s 16s Time ( ) 2 Fig. 10(c). Plot of mean ⎡⎣v 2 (t ) ⎤⎦ = mean ⎡ sin(2π t ) + sin(2 2π t ) ⎤ as a function of ⎢⎣ ⎥⎦ observation length (time) of the mean calculation. 16 the Technology Interface/Fall 2007 Cartwright E. RMS Value of a Triangular Waveform The rms value of a triangular waveform is known from calculus to be Vm / 3 = 0.5774Vm , and is of benefit in finding the level of ripple in a power supply. Here, we show this same result without using calculus. A graph of a triangular waveform is shown below in Fig. 11(a) and its square is given in Fig. 11(b). Note that the mean of this squared waveform is the same whether the mean is computed over the full cycle or over just half the cycle. Hence, we will compute the mean T T from − to . Note also that the equation of the triangular waveform over this time is 4 4 given by v(t ) = 4 Vm t T for − T T ≤t≤ . 4 4 (18) Therefore, the squared waveform has the equation v 2 (t ) = T T 16 2 2 Vm t for − ≤ t ≤ . 2 4 4 T (19) Voltage (V) Vm 0 -Vm -T/2 -T/4 0 Time (s) T/4 Fig. 11(a). Plot of a triangular waveform over a complete cycle. 17 T/2 the Technology Interface/Fall 2007 A B Square of Triangular Waveform (V2) Vm*Vm Cartwright 0 -T/2 C E -T/4 0 Time (s) D T/4 T/2 Fig. 11(b). Plot of the square of the triangular waveform above, over a complete cycle. Furthermore, the squared waveform over this time forms the parabola ACB, whose height is Vm2 and whose base is T / 2 . Therefore, the enclosed area of this parabola is well-known 2T 2 to be two-thirds the base times the height or Vm . However, the area under the 32 squared waveform is the enclosed area of the parabola ACB subtracted from the enclosing rectangle ABDE, whose base is T / 2 and whose height is Vm2 , and hence whose 1T 2 T area is Vm2 . So, the area below the curve of the squared triangular waveform is Vm , 2 32 1 and therefore its mean is this value divided by T / 2 or Vm2 . Taking the square root of 3 this gives the desired result. F. RMS Value of a Pulse Train We have left perhaps the easiest waveform for last. This waveform is a pulse train and is shown in Fig. 12(a), whereas its square is shown in Fig. 12(b). Clearly, the mean of the square is just the area of the squared voltage ( Vm2τ ) divided by the period, or Vm2τ / T . Hence, the rms value is given by Vm τ T , which is a well-known result from calculus. 18 the Technology Interface/Fall 2007 Cartwright Voltage (V) Vm 0 τ 0 T/2 Time (s) T Fig. 12(a). Plot of a pulse train over a complete cycle. Square of Voltage (V2) Vm*Vm 0 0 τ T/2 Time (s) T Fig. 12(b). Plot of the square of the pulse train above, over a complete cycle. 19 the Technology Interface/Fall 2007 Cartwright III. Conclusion In this paper, the rms value of various waveforms was determined without using calculus. This should be of immense help to the Technology student. Furthermore, a discussion of the importance of using the correct observation length to compute the mean was also given. References [1] Guillermo Rico, “Tech Tip: Effective or RMS Voltage of a Sinusoid,” the Technology Interface, Spring 2006, Vol. 6 No. 1 ISSN 1523-9926 http://engr.nmsu.edu/~etti/Spring06/ . [2] Boylestad, R. L., Introductory Circuit Analysis, 11th edition, Prentice Hall, Upper Saddle River, NJ, 2007. 20